# §25.10 Zeros

## §25.10(i) Distribution

The product representation (25.2.11) implies $\zeta\left(s\right)\neq 0$ for $\Re s>1$. Also, $\zeta\left(s\right)\neq 0$ for $\Re s=1$, a property first established in Hadamard (1896) and de la Vallée Poussin (1896a, b) in the proof of the prime number theorem (25.16.3). The functional equation (25.4.1) implies $\zeta\left(-2n\right)=0$ for $n=1,2,3,\dots$. These are called the trivial zeros. Except for the trivial zeros, $\zeta\left(s\right)\neq 0$ for $\Re s\leq 0$. In the region $0<\Re s<1$, called the critical strip, $\zeta\left(s\right)$ has infinitely many zeros, distributed symmetrically about the real axis and about the critical line $\Re s=\frac{1}{2}$. The Riemann hypothesis states that all nontrivial zeros lie on this line.

Calculations relating to the zeros on the critical line make use of the real-valued function

 25.10.1 $Z(t)=\exp\left(i\vartheta(t)\right)\zeta\left(\tfrac{1}{2}+it\right),$ ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\exp\NVar{z}$: exponential function, $Z(t)$: zeros function and $\vartheta(t)$: function Permalink: http://dlmf.nist.gov/25.10.E1 Encodings: TeX, pMML, png See also: Annotations for 25.10(i), 25.10 and 25

where

 25.10.2 $\vartheta(t)\equiv\operatorname{ph}\Gamma\left(\tfrac{1}{4}+\tfrac{1}{2}it% \right)-\tfrac{1}{2}t\ln\pi$

is chosen to make $Z(t)$ real, and $\operatorname{ph}\Gamma\left(\frac{1}{4}+\frac{1}{2}it\right)$ assumes its principal value. Because $|Z(t)|=|\zeta\left(\frac{1}{2}+it\right)|$, $Z(t)$ vanishes at the zeros of $\zeta\left(\frac{1}{2}+it\right)$, which can be separated by observing sign changes of $Z(t)$. Because $Z(t)$ changes sign infinitely often, $\zeta\left(\frac{1}{2}+it\right)$ has infinitely many zeros with $t$ real.

## §25.10(ii) Riemann–Siegel Formula

Riemann developed a method for counting the total number $N(T)$ of zeros of $\zeta\left(s\right)$ in that portion of the critical strip with $0. By comparing $N(T)$ with the number of sign changes of $Z(t)$ we can decide whether $\zeta\left(s\right)$ has any zeros off the line in this region. Sign changes of $Z(t)$ are determined by multiplying (25.9.3) by $\exp\left(i\vartheta(t)\right)$ to obtain the Riemann–Siegel formula:

 25.10.3 $Z(t)=2\sum_{n=1}^{m}\frac{\cos\left(\vartheta(t)-t\ln n\right)}{n^{1/2}}+R(t),$

where $R(t)=O\left(t^{-1/4}\right)$ as $t\to\infty$.

The error term $R(t)$ can be expressed as an asymptotic series that begins

 25.10.4 $R(t)=(-1)^{m-1}\left(\frac{2\pi}{t}\right)^{1/4}\frac{\cos\left(t-(2m+1)\sqrt{% 2\pi t}-\frac{1}{8}\pi\right)}{\cos\left(\sqrt{2\pi t}\right)}+O\left(t^{-3/4}% \right).$

Riemann also developed a technique for determining further terms. Calculations based on the Riemann–Siegel formula reveal that the first ten billion zeros of $\zeta\left(s\right)$ in the critical strip are on the critical line (van de Lune et al. (1986)). More than one-third of all the zeros in the critical strip lie on the critical line (Levinson (1974)).

For further information on the Riemann–Siegel expansion see Berry (1995).